where \(f(x)\)  is an arbitrary function. In the free theory \(Q_0(f)\)  is also a conserved quantity because the number operators \(a^{\dagger }a, b^{\dagger }b\)  are themselves conserved. This gives an infinite number of conserved charges corresponding to each function \(f(x)\)  [1]}. It acts on the creation operators as,
\({\begin{array}{c}e^{iQ_{0}(f)} a^{\dagger }(\omega ,x) e^{-iQ_{0}(f)} = e^{i e f(x)} a^{\dagger } (\omega ,x), \quad e^{iQ_0(f)} {\omega ,x,e} = e^{ie f(x)} {\omega ,x,e} \\e^{iQ_{0}(f)} b^{\dagger }(\omega ,x) e^{-iQ_{0}(f)} = e^{-ie f(x)} b^{\dagger } (\omega ,x), \quad e^{iQ_0(f)} {\omega ,x,-e} = e^{-ie f(x)} {\omega ,x,-e}\end{array}}\) 
